A line meets the coordinate axes at A and B. If the perpendicular distances from A and B to the tangent drawn at the origin to the circumcircle of $\mathrm{△}OAB$ are m and n respectively, then the diameter of that circle is.

Let $A=(a,0),B=(0,b)$

Equation of circumcircle of $\mathrm{\Delta }OAB\text{ is }(x-a)x+y(y-b)=0\Rightarrow x^2+y^2-ax-by=0$

Equation of tangent at  $(0,0)\text{ is }0+0-\frac{a}{2}(x+0)-\frac{b}{2}(y+0)=0\Rightarrow ax+by=0$ 

m = perpendicular distance from A to the tangent

n= perpendicular distance from B to the tangent $=\frac{b^2}{\sqrt{a^2+b^2}}$

$m+n==\frac{a^2}{\sqrt{a^2+b^2}}+\frac{b^2}{\sqrt{a^2+b^2}}=\sqrt{a^2+b^2}=AB.\therefore \text{ circle }=AB=m+n$